Let ${(x_1,x_2,z_1,z_2)}$ be real-valued vectors of equal length with
$${\hat{r}_p(x_1,x_2) \approx \hat{r}_p(z_1,z_2) > 0}$$
$${\hat{r}_p(x_1,z_1) \approx \hat{r}_p(x_2,z_2) > 0}$$
where $\hat{r}_p$ denotes the estimated Pearson correlation coefficient. Now perform two simple linear regressions
$$x_1 = \hat\beta_{0,1}+\hat\beta_{1,1}z_{1}+\hat{e}_1$$ $$x_2 = \hat\beta_{0,2}+\hat\beta_{1,2}z_{2}+\hat{e}_2$$
(The coefficients have two subscripts to make clear that these two regression equations do not share their coefficients.)
The purpose of the regressions is to extract the residuals $\hat{e}_i$ as versions of the $x_i$ adjusted for/uncorrelated with the corresponding $z_i$.
My question is whether general statements can be made about the magnitude of ${\hat{r}_p(\hat{e}_1,\hat{e}_2)}$ in relation to that of ${\hat{r}_p(x_1,x_2)}$, that is, is the estimated correlation between the $x_i$ expected to grow or shrink by "adjusting for" (removing linear correlation with) the $z_i$?
